Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{z^2 - 4z - 21}{z^2 - 7z}$
First factor the expressions in the numerator and denominator. $ \dfrac{z^2 - 4z - 21}{z^2 - 7z} = \dfrac{(z + 3)(z - 7)}{(z)(z - 7)} $ Notice that the term $(z - 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z - 7)$ gives: $q = \dfrac{z + 3}{z}$ Since we divided by $(z - 7)$, $z \neq 7$. $q = \dfrac{z + 3}{z}; \space z \neq 7$